Defined the loss, now we’ll have preciso compute its gradient respect to the output neurons of the CNN mediante order sicuro backpropagate it through the net and optimize the defined loss function tuning the net parameters. The loss terms coming from the negative classes are niente. However, the loss gradient respect those negative classes is not cancelled, since the Softmax of the positive class also depends on the negative classes scores.
The gradient expression will be the same for all \(C\) except for the ground truth class \(C_p\), because the punteggio of \(C_p\) (\(s_p\)) is con the nominator.
- Caffe: SoftmaxWithLoss Layer. Is limited esatto multi-class classification.
- Pytorch: CrossEntropyLoss. Is limited sicuro multi-class classification.
- TensorFlow: softmax_cross_entropy. Is limited puro multi-class classification.
Durante this Facebook rete informatica they claim that, despite being counter-intuitive, Categorical Ciclocross-Entropy loss, or Softmax loss worked better than Binary Cross-Entropy loss mediante their multi-label classification problem.
> supporto ferzu Skip this part if you are not interested mediante Facebook or me using Softmax Loss for multi-label classification, which is not standard.
When Softmax loss is used is verso multi-label contesto, the gradients get verso bit more complex, since the loss contains an element for each positive class. Consider \(M\) are the positive classes of verso sample. The CE Loss with Softmax activations would be:
Where each \(s_p\) in \(M\) is the CNN conteggio for each positive class. As mediante Facebook paper, I introduce per scaling factor \(1/M\) onesto make the loss invariant puro the number of positive classes, which ple.
As Caffe Softmax with Loss layer nor Multinomial Logistic Loss Layer accept multi-label targets, I implemented my own PyCaffe Softmax loss layer, following the specifications of the Facebook paper. Caffe python layers let’s us easily customize the operations done durante the forward and backward passes of the layer:
Forward pass: Loss computation
We first compute Softmax activations for each class and store them per probs. Then we compute the loss for each image per the batch considering there might be more than one positive label. We use an scale_factor (\(M\)) and we also multiply losses by the labels, which can be binary or real numbers, so they can be used for instance esatto introduce class balancing. The batch loss will be the mean loss of the elements mediante the batch. We then save the scadenza_loss sicuro display it and the probs onesto use them durante the backward pass.
Backward pass: Gradients computation
Sopra the backward pass we need to compute the gradients of each element of the batch respect preciso each one of the classes scores \(s\). As the gradient for all the classes \(C\) except positive classes \(M\) is equal preciso probs, we assign probs values to sbocco. For the positive classes con \(M\) we subtract 1 preciso the corresponding probs value and use scale_factor onesto competizione the gradient expression. We compute the mean gradients of all the batch onesto run the backpropagation.
Binary Cross-Entropy Loss
Also called Sigmoid Ciclocross-Entropy loss. It is verso Sigmoid activation plus a Ciclocross-Entropy loss. Unlike Softmax loss it is independent for each vector component (class), meaning that the loss computed for every CNN output vector component is not affected by other component values. That’s why it is used for multi-label classification, were the insight of an element belonging sicuro a indivisible class should not influence the decision for another class. It’s called Binary Cross-Entropy Loss because it sets up per binary classification problem between \(C’ = 2\) classes for every class con \(C\), as explained above. So when using this Loss, the formulation of Ciclocross Entroypy Loss for binary problems is often used: